Line integrals Definition and 127 Threads

Hey guys and gals, this isn't actually an assignment of any sort, so I didn't want to put it in the homework section. This is also my first post, though I have been lurking for quite a while, reading the copious amounts of information available here. :p Anyhow, could somebody please elaborate...

Homework Statement Let C be the boundary of the region bounded by the curves y=x^{2} and y=x. Assuming C is oriented counter clockwise, Use green's theorem to evaluate the following line integrals (a) \oint(6xy-y^2)dx and (b) \oint(6xy-y^2)dyHomework Equations The Attempt at a Solution...

State the Fundamental Theorem: Let F be a vector field. If there exists a function f such that F = grad f, then \int_{C} F \cdot dr = f(Q) - f(P) where P and Q are endpoints of curve C. _________________________________ I didn't receive any credit for this answer. Admittedly...

Vector field F(bar)= <6x+2y,2x+5y> fx(x,y)= 6x+2y fy(x,y)= 2x+5y f(x,y)= 3x^2+2xy+g(y) fy(x,y)=2x+g'(y) 2x+g'(y)= 2x+5y g'(y)= 5y g(y)= 5/2*y^2 f(x,y)=3x^2+2xy+(5/2)y^2 Then find the \int F(bar)*dr(bar) along curve C t^2i+t^3j, 0<t<1 I'm stuck on finding the last part for the F(bar)...

other than the physics (work) what are the applications of line integral? particularly does it have any use in finance or economics?

Homework Statement Suppose that p and q are points in U, where U is an open, path-connected, simply connected subset of Rn and c1 and c2 are smooth curves in Rn with c1(0)=c2(0)=p, c1(1)=c2(1)=q. Let w be a 1-form on U. Prove that the line integral of w over c1 equals the line integral of w...

Folks, 1) If we have \int F \cdot dr that is independent of the path, does that mean that the integral will always be 0? 2) For 2 dimensional problems when we evaluate line integrals directly and use Greens Theorem for every piece wise smooth closed curves C, arent we always calculating...

Homework Statement Evaluate this integral directly Homework Equations \int cos x sin y dx +sin x cos y dy on vertices (0,0), (3,3) and (0,3) for a triangle The Attempt at a Solution Does this have to evaluated parametrically using r(t)=(1-t)r_0+tr_1 for 0 \le t\le 1 or can I just...

Hi, I'm studying calculus 3 and am currently learning about conservative vector fields. ============================= Fundamental Theorem for Line Integrals ============================= Let F be a a continuous vector field on an open connected region R in ℝ^{2} (or D in ℝ^{3}). There exists...

Homework Statement There is a circle of equation x^2+y^2=1 and a vector field F (x; y) =< y + .5x, x + .3y >. Imagine the field zoomed in extremely close at (0,1), to the point where it looks like a constant field of <-1,.3>. Calculate the work from say (0,1) to (-.001, 1). The constant field...

I am just trying to get the conceptual basics in my head. Can I think of things this way... If you are taking the integral of a function f(z) along a curve γ in a region A. If the curve is closed and f(z) is analytic on the entire curve as well as everywhere inside the curve, then the...

hi experts as far I know the stokes theorem relates surface integral to line integral - but i am confuse how surface integral if represent area gets equal to length as represented by line integral.

Homework Statement Demonstrate that \int_{-\gamma} f(z)|dz|=\int_{\gamma} f(z)|dz| where \gamma is a piecewise smooth path and f is a function that is continuous on |\gamma|. Homework Equations The Attempt at a Solution This proof seems like it should be very simple, but I am...

Homework Statement A sleeve of mass m is constrained to move without friction along the x-axis. The sleeve is connected to the point (0, 2) on the y-axis by a spring as shown in the diagram below. Assume that Hooke’s “Law” is a good approximation for the restoring force exerted by the...

in line integrals we always need a vector element of distance. I can't understand the difference between ds and dr. is ds for all kinds of paths (even curly ones) and dr only for straight lines, or theyre the same? I am confused, or maybe dr is just the magnitude of ds, and the vector here is...

Homework Statement \int_{C}(x+yz)dx + 2xdy + xyzdz C goes from (1,0,1) to (2,3,1) and (2,3,1) to (2,5,2) The Attempt at a Solution For C going from (1,0,1) to (2,3,1) x=1+t, y=3t, z=1; 0\leq t \leq 1 x'(t)=1, y'(t)=3, z'(t)=0 \int^{1}_{0}(1+t+3t)*1dt + 2(1+t)*3dt + 0...

Homework Statement I have to calculate the following line integral \int_{\gamma}y^{2}cos(xy^{2})dx + 2xycos(xy^{2})dy where \gamma is the path defined by the equations x(t) = t^{4} and y(t)=sin^{3}(\frac{t\pi}{2}) t between 0 and 1Homework Equations Now I know that the formula for calculating...

If someone could link me to a tutorial on how to put in functions into a post, I would appreciate it, thanks. I am going to be putting in screen shots. Homework Statement http://img864.imageshack.us/img864/1517/scr1305133657.png" http://img864.imageshack.us/img864/1517/scr1305133657.png...

Homework Statement Given \mathbf{F} = \nabla f\; where \;f(x,y) = sin(x-2y) Find a curve C that is not closed and satisfy the equation \int_C \mathbf{F}\cdot dr = 0The Attempt at a Solution \nabla f = \;<cos(x - 2y),-2cos(x-2y)> So to satisfy the dot product being 0 (I am hoping I can do...

We are given a vector field: F=\frac{-y}{x^2+y^2} , \frac{x}{x^2+y^2} Then asked if F is conservative on R2 \ (0,0). I just solved the partial derivatives of each part of the vector field and they did indeed equal each other, but I don't under stand what the "\(0,0)" part means. We are then...

Homework Statement Compute the following line integral: \int_{\gamma} |z|^2 dz where \gamma(t) is the line segment from 2 to 3 + i Homework Equations Green's Theorem The Attempt at a Solution I originally started by saying that y = x - 2 and subing that into the equation "x^2 + y^2"...

Homework Statement what is the difference between path integral and line integral? Homework Equations n/a The Attempt at a Solution is path integral over a scalar function and line integral is over vector function? I'm confused about this pls help me understand thanks...

show that the line integral is indpendant of path and evaluate the integral on interval (0,1),(1,2) int c 1-ye^{-x}dx+e^{-x}dy can somone show me the procedure here looks like they just integrated 1-ye^(-x) on x to get 2/e I get a diffrent answer if I integrate e^(-x) on y same interval do I...

Homework Statement A wire lies along the piecewise linear curve extending from the point (4,3) to the point (6,15) to the point (12,15). If the density of the wire is given by (xy)=3xy+2y, use a line integral to find the mass of the wire. Homework Equations The Attempt at a Solution...

Conservative vector fields and line integrals Homework Statement A particle is subject to a force F defined by F\left( x,y \right)=\left(\begin{array}{c} y^{2} \\ 2xy \end{array}\right). The particle moves in a straight line C from (-1,2) to (1,3).[a] Calculate the work done by the force F as...

Homework Statement a) Use the fundamental theorem of line integrals to evaluate the line integral: ∫(2x/(x^2+y^2)^2)dx+(2y/(x^2+y^2)^2)dy (over C) Where C is the arc of the circle (x-4)^2+(y-5)^2=25 taken clockwise from (7,9) to (0,2). Explain why the fundamental theorem can be applied. b)...

Homework Statement Find the simple closed integral of (x+xy-y)(dx+dy) counterclockwise around the path of straight line segments from the origin to (0,1) to (1,0) to the origin... a)as a line integral b)using green's theorem Homework Equations Eq of line segment r(t)=(1-t)r0+tr1 Greens...

Alright, I have a conceptual question regarding Green's Theorem that I'm hoping someone here can explain. We recently learned in my college class that, by Green's Theorem, if C is a positively-oriented, piecewise-smooth, simple closed curve in the plane and D is the region bounded by C, then the...

Homework Statement Evaluate the line integral I = (x2z + yzexy) dx + xzexy dy + exy dz where C is the arc of the ellipse r(t) = (cost,sint,2−sint) for 0 <= t <= PI. [Hint: Do not compute this integral directly. Find a suitable surface S such that C is a part of the boundary ∂S and use...

Homework Statement A thin wire has the shape of the first quadrant part of the circle with center at the origin and radius a. If the density function is rho(x,y)=kxy, find the mass and center of mass of the wire. Homework Equations My parametric equation of the circle was x=a*cos(t) and...

If I draw a random curve over a scalar field, then it is not generally true that the line integral of the scalar field over the curve equals the difference between the value of the antiderivatives of the scalar field at the beginning and finishing points of the curve, as one can clearly see by...

Homework Statement I have a vector valued function that I need to integrate over a curve C (which I know how to do). I need to create a vector valued function r(t) for any position on the curve C (see the picture). r(t) is a defined area in the XY-plane and I'm pretty sure it needs a...

Homework Statement F = < z^2/x, z^2/y, 2zlog(xy)> F = \nabla f, where f = z^2log(xy) Homework Equations Evaluate \int F \cdot ds for any path c from P = (1/2, 4, 2) to Q = (2, 3, 3) contained in the region x > 0, y > 0, z > 0 Why is it necessary to specify that the path lie in the...

I saw somewhere triple line integrals, 3 integrals with a circle(the symbol) , may you tell me exactly what are called, to find & study them ?

Homework Statement F = (3x2 + 2y cos(xy))i + (2y + 2x cos(xy))j a - show that F is a gradient field b - calculate the integral of F dot dr where c includes the points -2,0 and 2,0 c - determine the value of the integral of F dot dr where c is any curve joining -2,0 and 2,0...

Homework Statement \int _c{(x^2 + y + \sqrt{x})dx + (y - x^2 + \sin{y}) dy Homework Equations The Attempt at a Solution I'm not sure which theorem to use here. Do I use Green's or the Divergence? Even once I get past this I'm not sure that I can get started.

Homework Statement In each part, evaluate the integral \int(3x+2y)dx+(2x-y)dy (A) The line segment from (0,0) to (1,1). (b) The parabolic arc y=x^2 (c) The curve y=sin(pi(x)/2) from (0,0) to (1,1) (D) The curve x=y^3 from (0,0) to (1,1). Homework Equations \int...

Finding line integrals -- please help! Given F = y/(x^2 + y^2) i - x / (x^2 + y^2) j Find the line integral of the tangential component of F from (-1,0) to (0,1) to (1,1) to (1,0) (assuming F is NOT path independent). --- I tried parameterizing each of the three paths using the formula r(t)...

When I take the line integral around a square shape path "C" as follows: From A to B to C to D to A C1 = A(0, 0) to B (4, 0) t i 0 <= t <= 4 C2 = B (4, 0) to C (4, 7) 4 i + (t - 4) j 4 <= t <= 11 C3 = C (4, 7) to D (0, 7) (15 - t) i + 7 j 11 <= t <= 15 C4 = D (0, 7)...

Homework Statement Hello, I'm writing a summary of all calculus I've learned during this term and now I'm on line integrals. I wrote this so far: http://img17.imageshack.us/img17/9776/algebrac.jpg But I have Sigma of x and y (a similar expression was in my lecture notes), but there is no dy...

If a question says something like: "evaluate \int(x*z*y)dx - (x-y)dy + (x^3)dz from (1,0,0,) to (1,0,2pi) along the curve (x,y,z)=(cos(t),sin(t),t)" or something like that, this is just basically splitting up a line integral? In my example, it would be the same as: \intcos(t)*t*sin(t)) *...

Line Integrals (yayyy!) Homework Statement Okay, so I have already done it using the surface integral; now I need to compute the 3 individual line integrals. By definition, the integral (I will call it I since I am that creative) is given by: I=\oint v\cdot\, dl v=<xy, 2yz...

Homework Statement Calculate \intF dr if C = C1 + C2 where C1 is the line segment from P1(-1,pi,-1) to P2(0,0,0) and C2 is the line segment from P2(0,0,0) to P3(2,0,4) vectorF= yz i + (xz - e^(z)siny) j + (e^(z)cosy + xy) k The Attempt at a Solution Im having problems setting up the...

Evaluate the line integral \int y^(2) dx + xy dy from A(1,0) to B(-1,4) with C: x = 1-t, y = t^(2), 0≤t≤2 I used: Do I make the limits from 0 to 2? What do I do with the A(1,0) and B(-1,4)? Please help? Thanks.

Hi, I am currently studying complex numbers and I am at the part we have to find line integrals over a simple closed curve gamma(t).. I know the definition, but when i read a problem I am n ot sure how to parameratize the curve. I was wondering if there are some tricks to this. Would...

Hi, I am currently studying complex numbers and I am at the part we have to find line integrals over a simple closed curve gamma(t).. I know the definition, but when i read a problem I am n ot sure how to parameratize the curve. I was wondering if there are some tricks to this. Would...

Homework Statement i) Define a path \gamma whose image is the ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 traced counterclockwise. ii) Show that \int \frac{1}{z} dz = \int \frac{1}{z} dz for a suitable circle \beta (NOTE: THE FIRST INTEGRAL IS OVER THE ELLIPSE \gamma, THE SECOND ONE IS...

Homework Statement I'm trying to set up a couple integrals. Suppose you have a closed loop and you want to find its area. You don't know what the shape of the loop is. All you know is that the length of the loop is L. The Attempt at a Solution These are my integrals. I just...

*Please don't just read the first post and leave. I'm adding to the thread with each attempt. I'm not asking someone to do it for me!* I guess this is a lousy first post, and I apologize, but I am desperate and would appreciate all the help I can get more than anyone could imagine! This is...

Hello, I am trying to solve for the surface area of a odd surface for a fire relief PSV and needed to do a line integral but I was reading into my calculus book and going back to the definition of arc length I am confused: L = lim_{n\rightarrow\infty}\sum (P_{i-1}*P_{i}) Multiplication should...